Recall that Case 1 corresponds to the situation when the acting group \(H\)  is contained in the parabolic \(Q\)  of \(G\) . As explained in §REF , it follows that \(H\)  fixes a point in \(G/Q\)  and hence stabilizes the fibre above the fixed point. Therefore, up to conjugating \(Q\) , we are left with studying the associated \(\mu \) -random walk on the fibre \(S/\Lambda \) , where the probability measure \(\mu \)  is seen as a Zariski-dense measure in a copy of \(\operatorname{SL}_2(\mathbb {R})\)  in the subgroup of \(S\) . This is then a particular situation of the setting treated in Benoist–Quint's work [1]}, [2]}. Consequently, the corresponding equidistribution results apply. We do not state the result here as it would be a repetition. We refer the reader to the more recent [3]}, where the compact support assumption of [2]} is relaxed to finite exponential moment.
